Geometry of Structural Transformations

Yet another example of the geometry of deformation of interest to the present enterprise is that of structural transformation. As was evidenced in chap. 1 in our discussion of phase diagrams, material systems admit of a host of different structural competitors as various control parameters such as the temperature, the pressure and the composition are varied. Many of these transformations can be viewed from a kinematic perspective with the different structural states connected by a deformation pathway in the space of deformation gradients. In some instances, it is appropriate to consider the undeformed and transformed crystals as being linked by an affine transformation. A crystal is built up through the repetition 

There are a number of different examples within which it is possible to describe the kinematics of structural transformation. Perhaps the simplest such example is that of the transformation between a cubic parent phase and a transformed phase of lower symmetry such as a tetragonal structure. We note that we will return to precisely such structural transformations in the context of martensitic microstructures in chap. 10. If we make the simplifying assumption that the transformed axes correspond with those of the parent phase, then the deformation mapping is of the form 

This corresponds in turn to a deformation gradient tensor of the form 

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